Sampling theorems on locally compact groups from oscillation estimates

Abstract

We present a general approach to derive sampling theorems on locally compact groups from oscillation estimates. We focus on the L 2 -stability of the sampling operator by using notions from frame theory. This approach yields particularly simple and transparent reconstruction procedures. We then apply these methods to the discretization of discrete series representations and to Paley-Wiener spaces on stratified Lie groups.

Sampling in Paley-Wiener spaces on combinatorial graphs

“…Some of the ideas and methods of this theory were recently extended to the cases of Riemannian manifolds, symmetric spaces, groups, and quantum graphs [7], [8], [10], [11], [14], [15], [25]- [32].…”

Paley‐Wiener estimates for the Heisenberg group

“…Finally, a word concerning the initial motivation for this paper: The interest in studying versions of the Paley-Wiener theorem for the Heisenberg group was sparked by sampling theory. The sampling results obtained in [7] provide asymptotic estimates only; in particular, there is no explicitly known critical sampling density. In view of the success of complex analysis methods in treating the analogous question for R, a Paley-Wiener theorem appeared to be a useful result.…”

“…(1.2) This theorem has many uses, in particular in sampling theory, where it allows a precise analysis of irregular sampling sets and critical densities. The space P W (G), for a stratified Lie group G, was introduced by Pesenson [21] via the spectral decomposition of a sub-Laplacian associated to G. The initial motivation for the definition of this space was to derive sampling theorems for Paley-Wiener functions, see also [7]. In [21], Pesenson further proved an analog of the Paley-Wiener theorem by showing that the elements of P W (G) could be characterized as entire vectors for the one-parameter group generated by the sub-Laplacian, satisfying a growth condition for the norm.…”

On the approximate form of Kluvánek's theorem

“…By Lemma 8, for each w > 0, / w , where w is given by (21), is a compact quotient group with transversal w given by (22). Let > 0 be given.…”

“…In a substantial paper [22], Führ and Gröchenig discuss sampling theorems in the difficult general case of non-abelian locally compact groups (see also [30]). They analyse sampling in reproducing kernel spaces using ideas from functional analysis, obtaining essentially a sampling theorem for wavelets and another for nilpotent Lie groups.…”